(i) If ` a + b + c = 5 , (a, b, c gt 0)`, find the greatest value of ` a^(3) b^(2) c^(4)` .
(ii) Find the greatest value of ` (5 - x)^(2) (x -2)^(2) ` , given ` 2 lt x lt 5 ` .

Explanation

(i) First a word about the solution . The AM -GM inequality has on the left sum of quantities
and on the right product of quantities . If sum happens to be constant , we can find the greatest value of the product . And if the product happnes to be constant , we can find the
least value of the sum .
As in the product ` a^(3) b^(2) c^(4)` , exponents of a, b, c are 3 , 2 and 4 respecitvely , we write .
`.....a + b + c = 3 ((a)/(3)) + 2((b)/(2)) + 4((c)/(4))`
Now , considering the 9 quantities `(a)/(3),(a)/(3),(a)/(3),(b)/(2), (b)/(2) , (c)/(4),(c)/(4),(c)/(4),(c)/(4)`
We have by AM-GM inequality.
`((a)/(3)+(a)/(3)+(a)/(3)+(b)/(2)+ (b)/(2) + (c)/(4)+(c)/(4)+(c)/(4)+(c)/(4))/(9) ge root(9)(((a)/(3))^(3)((b)/(2))^(2)((c)/(4))^(4))`
`rArr (a+b+c)/(9) ge root(9)((a^(3)b^(2)c^(4))/(3^(2)2^(2)4^(4)))`
`(5)/(9) ge root(9)((a^(3)b^(2)c^(4))/(3^(2)2^(2)4^(4)))`
`((5)/(9))^(9) ge (a^(3)b^(2)c^(4))/(3^(2)2^(2)4^(4))`
`rArr a^(3) b^(2)c^(4) ge ((5)/(9))^(9)3^(3)2^(2) 4^(4) = (5^(9) 2^(10))/(3^(15))`
Thus , the greatest value of ` a^(3) b^(2) c^(4) is (2^(10) 5^(9))/(3^(15))`
And this is attained at `(a)/(3) = (b)/(2) = (c)/(4) = (a + b+c)/(2+3+4) = (5)/(9)`
i.e., ` a = (5)/(3), b = (10)/(9) , c= (20)/(9)`
(ii) This problem is of the same variety as the previous one, except that it seems the sum is not specified .
Set ` 5 - x = , x=2 = b` , so that a, b `gt ` 0
Also a + b = 3
We need the greatest value of `a^(2) b^(5)`
Write ` a + b = 2 ((a)/(2)) + 5 ((b)/(5))`
Now consider the 7 quantities `(a)/(2), (a)/(2), (b)/(5),(b)/(5),(b)/(5),(b)/(5),(b)/(5)` We have by AM-GM inequality
`((a)/(2)+ (a)/(2)+ (b)/(5)+(b)/(5)+(b)/(5)+(b)/(5)+(b)/(5))/(7) ge root(7)(((a)/(2))^(2) ((b)/(5))^(5))`
`rArr (3)/(7) ge root(7)((a^(2) b^(5))/(2^(2) 5^(5)))`
`rArr((3)/(7))^(7) ge (a^(2) b^(5))/(2^(2) 5^(5))`
`rArr a^(2) b^(5) le (2^(2) 3^(7) 5^(5))/(7^(7))`
Thus the greatest value of ` a^(2) b^(5) is (2^(2)3^(7)5^(5))/(7^(7))`
And it is attained when ` (a)/(2) = (b)/(5) rArr (5-x)/(2) = (x-2)/(5) rArr x = (29)/(7)`